In this article, we introduce the idea of nonlinear Suzuki (F, R)-contractions, which is patterned after the contractive conditions due to Suzuki [Nonlinear Anal. 71 (2009) 5313-5317] and Wardowski [Fixed Point Theory Appl. (2012) 94:6]. Utilizing our newly introduced contraction, we establish some relation-theoretic fixed point theorems. Furthermore, we adopt an example to highlight the genuineness of our newly proved results. Finally, we use our main results to establish the existence and uniqueness of solution for a nonlinear matrix equation.